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Invariant Fields of Symplectic and Orthogonal Groups

The projective orthogonal and symplectic groups $PO_n(F)$ and $PSp_n(F)$ have a natural action on the $F$ vector space $V' = M_n(F) \oplus ... \oplus M_n(F)$. Here we assume $F$ is an infinite field of characteristic not 2. If we assume there is more than one summand in $V'$, then the invariant fields $F(V')^{PO_n}$ and $F(V')^{PSp_n}$ are natural objects. They are, for example, the centers of generic algebras with the appropriate kind of involution. This paper considers the rationality properties of these fields, in the case $1,2$ or 4 are the highest powers of 2 that divide $n$. We derive rationality when $n$ is odd, or when 2 is the highest power, and stable rationality when 4 is the highest power. In a companion paper [ST] joint with Tignol, we prove retract rationality when 8 is the highest power of 2 dividing $n$. Back in this paper, along the way, we consider two generic ways of forcing a Brauer class to be in the image of restriction.